(1) To Yearn

Originally (per FANCY II), a sort of vague, indeterminate yearning or tendency; sometimes the physical activity resulting therefrom. Over the years it has become less vague and more intense a reaction, to the point that cries of "Faunch! Faunch!" may be heard, indicating a high degree of desire for the subject of the reaction. The extreme form is Faunching Backwards.

(1) to desire, to want; to yearn for–with sexual overtones, “I have a serious faunch for Darryl Hannah“ or “I have a serious faunch for James Marsters”–or without, "We’ve got a serious faunch for chocolate."
(2) to hunt for, to acquire; "He's gone on the faunch for some bheer."

from Fancyclopedia 2 Supplement ca. 1960:
The curious history of this word in fandom deserves mention. "Faunch" first appeared in Pro-Phile in a review writ by Noel Loomis; he mentioned his "typewriter faunching at the platen while he finished reading a story". The word fascinated Boggs, who used it years later in a W03W letter; there it fascinated Grennell who used it in Grue and other places. The word actually refers to the way a horse champs at the bit, impatient to prance away as soon as the reins are loosened. This is known as faunching at the bit and was a common expression of yore. It has gone into obscurity, naturally, since the advent of the auto; fandom is probably one of the few places where it is still used.